# Lack of intuition for distribution function in micro and macro state description

I am a mathematician who is trying to understand statistical mechanics / thermodynamics. I need a hint wrt the interpretation / meaning of the distribution function. Currently I seem to have a basic misunderstanding which is a show stopper for further progress. My current understanding is as follows:

Micro state: I have a large number of particles ($$n$$) and a phase space $${\cal Q}$$ of the corresponding Hamiltonian system, i.e. $${\Bbb R}^{6n}$$. I consider points in this phase space and trajectories through it.

Macro state: I want to concentrate on more essential aspects of the system, such as descriptions in terms of energy, pressure, temperature etc. So, one macro state $$m$$ corresponds to a possibly very large set $$M$$ of micro states. In principle, given $$m$$, I could find $$M$$.

My Problem: Numerous texts now move ahead and introduce a distribution function or probability density on the set of micro states. I do not understand why I should be interested in considering a distribution function on the micro states.

Suppose, I have macro state $$m$$. So a micro state description could, in my understanding, already be given by the set $$M \subset {\cal Q}$$ of micro states. By the indifference principle I could assume that all of them are equally probable, which would give me some sort of distribution function / probability density. However, I read the texts in such a manner, that there could be several different distribution functions / probability densities. I want to understand which additional physical insight / intuition is modeled by the additional structure provided when moving from the set $$M$$ to a distribution function / probability density on the set.